Billy suggested that our intruder may be a lizard of some sort, which, for some strange reason, seems worse to me than a spider.
After thinking about it, he may be right. The trail it leaves behind isn't sticky and disintegrates when I touched it, like maybe it had been slimy (Ugh!) and dried out.
If it is a lizard, I may need to fire my cat. She's obviously not doing her job! It may be because she's old (nearly 15) and lazy, but I still see her run and jump and play from time to time. I think she's actually scared of it. She's no cat; she's a chicken!